Variational Inference (VI) refers to methods that produce an approximate distribution by minimizing some quantification of divergence between the approximation and the desired posterior distribution [1]}. For the purposes of this paper, we will use VI to refer to the most common flavor of variational methods, namely minimizing the Kullback-Leibler (\({\rm KL}\) ) divergence between an approximate distribution from a fixed family and the desired distribution [2]}, [3]}, [4]}. Whereas sampling methods can be thought of as taking place “in the space of random variables,” VI takes place “in the space of parameters.” For instance, samples of a real-valued scalar are themselves real-valued scalars, but the variational problem of finding the best-fitting Gaussian is an optimization problem in the two-dimensional space of the Gaussian's parameters (e.g. mean and variance). The best-fitting approximate distribution is often used directly as a proxy for the true posterior in subsequent calculations, which can greatly simplify those downstream calculations if the approximate distribution is itself easy to integrate. In contrast to MCMC, VI is often used in cases where speed is more important than asymptotic bias [5]}, [1]}.
